What is the integral of e^{-st}t ?

The integral of e^{-st}t is 1/(s^2)(-se^{-st}t-e^{-st})+C

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integral of e^{-st}t

\int e^{- s t} t d t

\frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

Solution steps

\int e^{- s t} t d t

Apply u-substitution

: \int \frac{e ^{u} u}{s ^{2}} d u

= \int \frac{e ^{u} u}{s ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{s ^{2}} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{s ^{2}} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{s ^{2}} (e^{u} u - e^{u})

Substitute back

u = - s t

= \frac{1}{s ^{2}} (e^{- s t} (- s t) - e^{- s t})

Simplify

= \frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t})

Add a constant to the solution

= \frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

f (x) = 2 x^{3} - x^{2} + 9, at x = 2

x \to \infty lim (- \frac{π}{2})

x \to 0 lim ((1 - \frac{1}{x})^{x^{2}})

\int \frac{1}{( 49 + x ^{2} ) ^{2}} d x

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{1}{z}})

What is the integral of e^{-st}t ?
The integral of e^{-st}t is 1/(s^2)(-se^{-st}t-e^{-st})+C